Exercises

1. Download Historical Data

   • Get the code for S & P 500 Index using the Yahoo Finance database either from the Quandl website or the R console.
   • Download historical close prices for the years 2010 -- 2014 and save them in the variable close_prices.

2. Visual Inspection of the Data

   • Plot the close prices as a linechart. What kind of process would be appropriate?
   • Plot the logarithm (log()) of the close prices as a linechart. What kind of process would be appropriate?
   • Calculate the log - returns and save your results in the variable log_returns.
   • Plot the log - returns. What kind of process would be appropriate?
   • Plot the acf acf() of the log - returns and the squared log - returns.

3. Fitting a GARCH Model to Historical Data

   • Use the garchFit() function to fit a GARCH(1,1) model to the log - returns.
   • Replace the normal distribution with a Student - t - distribution.
   • Which model explains the data better?

4. Forecasting Volatility (Homework)

   • Write your own function vola_forecast() for k - step ahead forecasts \(\sigma_T(k)\), with forecast origin \(T\), of the volatility for the two models from exercise 3.
   • Note:
     • You can check your results by comparing them with the output from the predict() function in package fGarch which returns values of \(\sigma_T(k)\) .
     • You can use the following forecasting relations for a GARCH(1,1) model: $$\begin{align*} \sigma^2_T(1) &= \omega + \alpha_1 \epsilon_T^2 + \beta_1 \sigma_T^2 \\ \\ \sigma^2_T(k) &= \dfrac{\omega (1 - (\alpha_1 + \beta_1)^{k-1})}{1 - \alpha_1 + \beta_1} + (\alpha_1 + \beta_1)^{k-1} \sigma^2_T(1) \end{align*}$$

vola_forecast <- function(ARGUMENTS) {

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}